Intended primarily to stimulate family summer travel, the new airfare

Hi parijit , option E is correct. Both E and C use elliptical construction,
but option C introduces the an unnecessary and ungrammatical phrase "that of."

• Option C contains a parallelism error because it uses the pronoun that to "make a copy" of
some previous noun.**
In option C, the end of the sentence is
. . . shortens the purchase requirement... to a minimum of seven days rather than that of fourteen.

THAN is the comparison marker
RHS: that of fourteen
LHS: a minimum of seven days

Those two items are not parallel.
The issue is subtle.
"That of" implies that something "belongs to" fourteen,
or that fourteen, the number, is an attribute of something.
(We can't just stick "days" on the end. Analyze in isolation from E.)
In C, quite literally, the word fourteen is being compared to minimum.
Nothing belongs to fourteen ("that of fourteen"). (Nothing belongs to seven, either.)
"Seven days" is an attribute of the minimum advance purchase requirement. Fourteen is not.

Be careful not to read the word days into C.
We already have one confusing noun form, namely, the phrase that of.
In the prompt, fourteen is not followed by days. The comparison in C illogically becomes
minimum of seven days . . . that of fourteen.

Those two items are not parallel.

• In option E, the end of the sentence means
.. shortens the ... purchase requirement ... to a minimum of seven days rather than fourteen [days].
The object of the preposition OF is both clear and close to "fourteen."
The object is "seven days." Seven days and fourteen [days - what else could it be?] are parallel as the objects of the preposition OF,
and they clearly belong to ONE noun, minimum.

• Really, the easiest test is to ask: does E create ambiguity?
Recall the classic example, I like Mary more than John.
Do I like Mary more than I like John? [If yes, then I should write, I like Mary more than I like John.]
Or do I like Mary more than John likes Mary? [if yes, I should write, I like Mary more than John does.]

(E) does not create ambiguity (and there is no verb tense shift involved,
in which case we would have to adjust).
There is no rule about omission of nouns except that in context, we may omit as long as the sentence is clear.
That rule depends on context.

It is possible to argue that the phrase "that of" refers to "minimum of" and that fourteen clearly parallels seven days.

In that case, we do not need the that of in (C).
From Manhattan SC Guide, 6th edition, page 160, in which the author asserts that GMAC sometimes inserts "that of" or "those of" as a trap:
Unnecessary words: The field I most enjoy studying is that of literature.
Better construction: The field I most enjoy studying is literature.
-- Literature IS a field.
-- Similarly, we have a minimum requirement [some number of days before we can do X]. The minimum has changed. One minimum IS/WAS "seven days," and the only other number mentioned is "fourteen."
We do not need THAT OF to tell us what fourteen refers to, i.e., days.

I hope that helps.

** THAT OF?
• Sometimes a sentence contains that as a pronoun to make a "copy" of something.
Wrong: The material tested in some sections of the GMAT is like the LSAT.
Like is a comparison word. In this case, the material (the content) is being compared to the actual LSAT test.
Correct: The material tested in some sections of the GMAT is like that of the LSAT.
By inserting a "copy" of "material tested in some sections", this version compares material to material.
To check whether something is sensible, we can always replace a pronoun with its noun.
The material tested in some sections of the GMAT is like the material tested in some sections of the LSAT.

• We often need "that of" or "those of" in characteristics / attributes (OF someone or something), or possession (of something by someone)
In elliptical construction, we typically need to use that to "make a copy"
-- when we compare an attribute (compare X of Y to X of Z)
Wrong: He liked the collection of fiction rather than essays.
Wrong: He liked the collection of fiction more than essays.
In both cases, collection is incorrectly compared to essays.
Correct: He liked the collection of fiction rather than THAT [the collection] of essays.

-- when we compare a noun in possessive case, such as
Wrong: I like the painting by Matisse more than Mondrian. [compares painting to the artist Mondrian]
Correct: I like the painting by Matisse more than that by Mondrian. [compares a painting to a painting]